Aim

Model Rainfall Extremes on an Australian Wide Scale

Challenge

Varied climate and complex topography

Clustering

Distance for clustering is based on extremal dependence.

Calculate our distance \(d(x, x+h)\) using the F-madogram (Cooley 2006):

\[ d(x, x+h) = \dfrac{1}{2} \mathbb{E} \left| \left[ F(Z(x)) - F(Z(x+h)) \right| \right] \] where \(F\) is the empirical distribution function of \(Z(x)\) the annual maximum rainfall at location \(x\) .

Pros: Non-parametric, fast and only use the raw data.

Can write our F-madogram as a function of the extremal coefficient, \(\theta(h)\) \[ d(x,x + h) = \dfrac{\theta(h) - 1}{2(\theta(h) + 1)}, \]

where \(\theta(h)\) is a measure of extremal dependence,

\[ \mathbb{P}\left( Z(x) \leq z, Z(x + h) \leq z \right) = \left[\mathbb{P}(Z(x)\leq z) \right]^{\theta(h)}, \quad x,h \in \mathbb{R}^2 \] where \(Z\) is unit Frechet.
(For full details refer to Bernard et al. 2013.)

Clustering - K-medoids

Model

Local level:
+ Fit a Max-stable process
+ Natural extension of univariate extreme value theory to extremes in continuous space with dependence

Classification - KNN

Questions?

This work has been supported by the Australian Research Council through the Laureate Fellowship FL130100039, CSIRO and ACEMS.